Measures of Central Tendency

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 Week 1 :Lecture 1
Measure of central tendency and its types:
Arithmetic mean for ungroup and group
data. Its uses and applications

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Measures of Central Tendency
A measure of central tendency is a
descriptive statistic that describes the
average, or typical value of a set of scores.

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 Types of Averages
There are five common measures of central
tendency.
Arithmetic Mean
Median
Mode
Geometric Mean
Harmonic Mean
First three known as primary and last two
known as secondary. 4
 The Mean
X

The mean is sum of all observations divide

by no of observations.
The mean of a population is represented by
the Greek letter ; the mean of a sample is
represented by X
the arithmetic average of all the scores
(X)/N

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 The Mean (Arithmetic Mean)
•It is the Arithmetic Average of data values:

x
n
 xi xi  x2      xn
i 1

Sample Mean
n n
•The Most Common Measure of Central Tendency

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6
 Calculating the Mean
Calculate the mean of the following data:
1 5 4 3 2
Sum the scores (X):
1 + 5 + 4 + 3 + 2 = 15
Divide the sum (X = 15) by the number of
scores (N = 5):
15 / 5 = 3
Mean = X = 3
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 Mean for group data
Hourly wages No of workers   xi fx

50-54 4  52 208

55-59 8  57 456

60-64 12   62 744

65-69 20   67 1340

70-74 16   72 1152

75-79 10   77 770

.80-84 5  82 410

Total 75     5080

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 Where x is mid point and calculte by using
formula
mid point is equal to Lower limit +upper limit
and divide by 2.

X

fx
5080
 67.733
 f 75

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 Properties of A.M
the number, m, that makes (X - m) equal to 0
the number, m, that makes (X - m)2 a
minimum
Draw back
Affected by Extreme Values (Outliers)

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 When To Use the Mean
You should use the mean when
the data are interval or ratio scaled
Many people will use the mean with ordinally scaled data
too
and the data are not skewed
The mean is preferred because it is sensitive to
every score
If you change one score in the data set, the mean
will change
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 Week 1:Lecture 2
Measure of central tendency and its types:
Median, quintiles for ungroup and group
data. Its uses and applications

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 The Median
The median is simply another name for the
50th percentile
It is the score in the middle; half of the scores
are larger than the median and half of the scores
are smaller than the median

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How To Calculate the Median
Conceptually, it is easy to calculate the
median
There are many minor problems that can occur;
it is best to let a computer do it
Sort the data from highest to lowest or
lowest to highest.
Find the score in the middle
middle = (n+ 1) / 2 th value
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• If n is odd, the median is the middle number.
• If n is even, the median is the average of the 2
middle numbers.

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 Median Example
What is the median of the following scores:
10 8 14 15 7 3 3 8 12 10 9
Sort the scores:
15 14 12 10 10 9 8 8 7 3 3
Determine the middle score:
middle = (n + 1) / 2 = (11 + 1) / 2 = 6
Middle score = median = 9
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 Median Example
What is the median of the following scores:
24 18 19 42 16 12
Sort the scores:
42 24 19 18 16 12
Determine the middle score:
middle = (N + 1) / 2 = (6 + 1) / 2 = 3.5
Median = average of 3rd and 4th scores:
(19 + 18) / 2 = 18.5
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 Median for group data

h n
Median  l  (  c)
f 2

L lower class limit of selected class

h class interval
f frequency of selected class
n total frequency
C preceding cummulative frequency 18
 When To Use the Median
The median is often used when the
distribution of scores is either positively or
negatively skewed
The few really large scores (positively skewed)
or really small scores (negatively skewed) will
not overly influence the median.
Median is not Affected by Extreme Values

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 Quintiles
Quartiles
Deciles
Percentiles

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 Quartiles
Not a Measure of Central Tendency
Split Ordered Data into 4 Quarters

25% 25% 25% 25%

Position of i-th
Q1 Quartile:
Q2 Q3 of point
position
i(n+1)
Qi  4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22

Position of Q1 = 1•(9 + 1) = 2.50 Q1 =12.5

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 Week 1:Lecture 3
Measure of central tendency and its types:
Mode for ungroup and group data. Its uses
and applications.
Relationship among Mean, median and
mode.

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 The Mode
The mode is the score 6

that occurs most 5

frequently in a set of 4

Frequency
data 3

0
75 80 85 90 95
Score on Exam 1

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 Bimodal Distributions
When a distribution 6

has two “modes,” it is 5

called bimodal 4

Frequency
3

0
75 80 85 90 95
Score on Exam 1

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 Multimodal Distributions
If a distribution has 6

more than 2 “modes,” 5

it is called multimodal 4

Frequency
3

0
75 80 85 90 95
Score on Exam 1

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 Mode for group data

f m  f1
mod e  l  h
( f m  f1 )  ( f m  f 2 )
L is lower class limit of modal class
fm maximum frequency
f1 preceeding frequency of modal class
f2 preceeding frequency of modal class
h class interval
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 For group Data
Hourly wages No of workers

50-54 4

55-59 8

60-64 12 f1
65-69 20 fm
70-74 16 f2
75-79 10

.80-84 5

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Calculte the mode by putting the
value in formula
(20  12)
Mode=65+ (20  12)  (20  16)  5

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 When To Use the Mode
The mode is not a very useful measure of central
tendency
It is insensitive to large changes in the data set
That is, two data sets that are very different from each other
can have the same mode
7 120

6 100

5
80
4
60
3
40
2

1 20

0 0 29
1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 100
 When To Use the Mode
The mode is primarily used with nominally
scaled data
It is the only measure of central tendency that is
appropriate for nominally scaled data

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 Relations Between the Measures
of Central Tendency
In symmetrical
distributions, the median
and mean are equal
For normal distributions,
mean = median = mode
In positively skewed
distributions, the mean is
greater than the median
and mode
In negatively skewed
distributions, the mean is
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smaller than the median &
 Shapes
Describes How Data Are Distributed
Measures of Shape:
Symmetric or skewed

Left-Skewed Symmetric Right-Skewed

Mean Median Mode Mean = Median = Mode Mode Median Mean
For practice Question see book “Statistical
Techniques in Business and Economics”.
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